High-Order Accurate Entropy Stable Finite Difference Schemes for One- and Two-Dimensional Special Relativistic Hydrodynamics

Abstract

This paper develops the high-order accurate entropy stable finite difference schemes for one-and two-dimensional special relativistic hydrodynamic equations. The schemes are built on the entropy conservative flux and the weighted essentially non-oscillatory (WENO) technique as well as explicit Runge-Kutta time discretization. The key is to technically construct the affordable entropy conservative flux of the semi-discrete second-order accurate entropy conservative schemes satisfying the semi-discrete entropy equality for the found convex entropy pair. As soon as the entropy conservative flux is derived, the dissipation term can be added to give the semidiscrete entropy stable schemes satisfying the semi-discrete entropy inequality with the given convex entropy function. The WENO reconstruction for the scaled entropy variables and the high-order explicit Runge-Kutta time discretization are implemented to obtain the fully-discrete high-order entropy stable schemes. Several numerical tests are conducted to validate the accuracy and the ability to capture discontinuities of our entropy stable schemes.

Gas kinetic flux solver based high-order finite-volume method for simulation of two-dimensional compressible flows

“…Apart from the flux solver, the high-order spatial discretization to the macroscopic governing equations is another important issue for the high-order scheme. The commonly used methods include high-order finite volume (FV) [34][35][36], highorder finite difference (FD) [37,38], discontinuous Galerkin (DG) [39,40], spectral volume (SV) [41], spectral difference (SD) [42], and the weighted essential nonoscillatory (WENO) scheme [43,44], etc. Among them, the high-order FV method can be extended from its second-order counterpart intuitively, which is widely used in practical engineering.…”

Genuinely multidimensional physical-constraints-preserving finite volume schemes for the special relativistic hydrodynamics

“…Such works were successfully extended to the special relativistic magnetohydrodynamics (RMHD) in [48,49], where the importance of divergence-free fields in achieving PCP methods is shown. Recently, the entropy-stable schemes were also developed for the special RHD or RMHD equations [12,14,15,16,7,17]. Most of the above mentioned methods are built on the 1D Riemann solver, which is used to solve the local 1D Riemann problem at the cell interface by picking up flow variations that are orthogonal to the cell interface and then give the exact or approximate Riemann solution.…”